R, 159 153 bytes

function(P,Q,a=h(P),b=h(Q))paste0(b[1,]%o%a[1,],"x^",outer(b[2,],a[2,],"+"),collapse=" + ")

h=function(s,`/`=strsplit)sapply(el(s/" \+ ")/"x\^",strtoi)

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I really wanted to use outer, so there's almost surely a more efficient approach.

Pyth - 39 bytes

LmsMcdK"x^"%2cb)j" + "m++*FhdKsedCM*FyM

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Haskell, 124 bytes

import Data.Lists

s=splitOn

z=map(map read.s"x^").s"+"

a#b=intercalate" + "[shows(u*p)"x^"++show(v+q)|[u,v]<-z a,[p,q]<-z b]

Note: TIO lacks Data.Lists, so I import Data.Lists.Split and Data.List: Try it online!

Ruby, 102 bytes

->a,b{a.scan(w=/(-?d+)x.(d+)/).product(b.scan w).map{|x,y|(eval"[%s*(z=%s;%s),z+%s]"%x+=y)*"x^"}*?+}

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JavaScript, 112 bytes

I found three alternatives with the same length. Call with currying syntax.

A=>B=>(P=x=>x.split` + `.map(x=>x.split`x^`))(A).flatMap(a=>P(B).map(b=>a[0]*b[0]+'x^'+(a[1]- -b[1]))).join` + `

f=

A=>B=>(P=x=>x.split` + `.map(x=>x.split`x^`))(A).flatMap(a=>P(B).map(b=>a[0]*b[0]+'x^'+(a[1]- -b[1]))).join` + `



console.log( f('5x^4')('3x^23') )

console.log( f('6x^2 + 7x^1 + -2x^0')('1x^2 + -2x^3') )

console.log( f('3x^1 + 5x^2 + 2x^4 + 3x^0')('3x^0') )

console.log( f('4x^3 + -2x^14 + 54x^28 + -4x^5')('-0x^7') )

console.log( f('4x^3 + -2x^4 + 0x^255 + -4x^5')('-3x^4 + 2x^2') )

A=>B=>(P=x=>x.split` + `.map(x=>x.split`x^`))(A).flatMap(([c,e])=>P(B).map(([C,E])=>c*C+'x^'+(e- -E))).join` + `

f=

A=>B=>(P=x=>x.split` + `.map(x=>x.split`x^`))(A).flatMap(([c,e])=>P(B).map(([C,E])=>c*C+'x^'+(e- -E))).join` + `



console.log( f('5x^4')('3x^23') )

console.log( f('6x^2 + 7x^1 + -2x^0')('1x^2 + -2x^3') )

console.log( f('3x^1 + 5x^2 + 2x^4 + 3x^0')('3x^0') )

console.log( f('4x^3 + -2x^14 + 54x^28 + -4x^5')('-0x^7') )

console.log( f('4x^3 + -2x^4 + 0x^255 + -4x^5')('-3x^4 + 2x^2') )

A=>B=>(P=x=>[...x.matchAll(/(S+)x.(S+)/g)])(A).flatMap(a=>P(B).map(b=>a[1]*b[1]+'x^'+(a[2]- -b[2]))).join` + `

f=

A=>B=>(P=x=>[...x.matchAll(/(S+)x.(S+)/g)])(A).flatMap(a=>P(B).map(b=>a[1]*b[1]+'x^'+(a[2]- -b[2]))).join` + `



console.log( f('5x^4')('3x^23') )

console.log( f('6x^2 + 7x^1 + -2x^0')('1x^2 + -2x^3') )

console.log( f('3x^1 + 5x^2 + 2x^4 + 3x^0')('3x^0') )

console.log( f('4x^3 + -2x^14 + 54x^28 + -4x^5')('-0x^7') )

console.log( f('4x^3 + -2x^4 + 0x^255 + -4x^5')('-3x^4 + 2x^2') )

Jelly, 28 bytes

ṣ”+ṣ”xV$€)p/ZPSƭ€j⁾x^Ʋ€j“ + 

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Full program. Takes the two polynomials as a list of two strings.

C# (Visual C# Interactive Compiler), 192 190 188 bytes

n=>m=>string.Join(g="+",from a in n.Split(g)from b in m.Split(g)select f(a.Split(p="x^")[0])*f(b.Split(p)[0])+p+(f(a.Split(p)[1])+f(b.Split(p)[1])));Func<string,int>f=int.Parse;string p,g;

Query syntax seems to be a byte shorter than method syntax.

At least I beat SNOBOL.

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SNOBOL4 (CSNOBOL4), 192 176 bytes

	P =INPUT

	Q =INPUT

	D =SPAN(-1234567890)

P	P D . K ARB D . W REM . P	:F(O)

	B =Q

B	B D . C ARB D . E REM . B	:F(P)

	O =O ' + ' K * C 'x^' W + E	:(B)

O	O ' + ' REM . OUTPUT

END

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	P =INPUT				;* read P

	Q =INPUT				;* read Q

	D =SPAN(-1234567890)			;* save PATTERN for Digits (or a - sign); equivalent to [0-9\-]+

P	P D . K ARB D . W REM . P	:F(O)	;* save the Koefficient and the poWer, saving the REMainder as P, or if no match, goto O

	B =Q					;* set B = Q

B	B D . C ARB D . E REM . B	:F(P)	;* save the Coefficient and the powEr, saving the REMainder as B, or if no match, goto P

	O =O ' + ' K * C 'x^' W + E	:(B)	;* accumulate the output

O	O ' + ' REM . OUTPUT			;* match ' + ' and OUTPUT the REMainder

END

Python 2, 130 bytes

lambda a,b:' + '.join([`v*V`+'x^'+`k+K`for V,K in g(a)for v,k in g(b)])

g=lambda s:[map(int,t.split('x^'))for t in s.split(' + ')]

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JavaScript (Babel Node), 118 bytes

Takes input as (a)(b).

a=>b=>(g=s=>[...s.matchAll(/(-?d+)x.(d+)/g)])(a).flatMap(([_,x,p])=>g(b).map(([_,X,P])=>x*X+'x^'+-(-p-P))).join` + `

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Haskell, 133 bytes

f""=

f t|[(a,_:_:u)]<-reads t,[(i,v)]<-reads u=(a,i):f(drop 3v)

p!q=drop 3$do(a,i)<-f p;(b,j)<-f q;" + "++shows(a*b)"x^"++show(i+j)

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f parses a polynomial from a string, ! multiplies two of them and formats the result.

Python 2, 193 bytes

import re

f=re.finditer

lambda a,b:' + '.join(' + '.join(`int(m.group(1))*int(n.group(1))`+'x^'+`int(m.group(2))+int(n.group(2))`for n in f('(-?d+)x^(d+)',b))for m in f('(-?d+)x^(d+)',a))

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Side note: First time doing a code golf challenge, so sorry if the attempt sucks haha

Retina, 110 bytes

SS+(?=.*n(.+))

 $1#$&

|" + "L$v` (-?)(d+)x.(d+).*?#(-?)(d+)x.(d+)

$1$4$.($2*$5*)x^$.($3*_$6*

--|-(0)

$1

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SS+(?=.*n(.+))

 $1#$&

Prefix each term in the first input with a #, a copy of the second input, and a space. This means that all of the terms in copies of the second input are preceded by a space and none of the terms from the first input are.

|" + "L$v` (-?)(d+)x.(d+).*?#(-?)(d+)x.(d+)

$1$4$.($2*$5*)x^$.($3*_$6*

Match all of the copies of terms in the second input and their corresponding term from the first input. Concatenate any - signs, multiply the coefficients, and add the indices. Finally join all of the resulting substitutions with the string  + .

--|-(0)

$1

Delete any pairs of -s and convert -0 to 0.

Perl 6, 114 bytes

{my&g=*.match(/(-?d+)x^(d+)/,:g)».caps».Map;join " + ",map {"{[*] $_»{0}}x^{[+] $_»{1}}"},(g($^a)X g $^b)}

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