To determine the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] that make the equation true, let's simplify the left-hand side of the given equation step-by-step.
First, we start with the expression:
[tex]\[
(5 x^7 y^2)(-4 x^4 y^5)
\][/tex]
Here, we'll break down the multiplication into coefficients and variables separately.
### Step 1: Multiplying the Coefficients
We look at the numerical coefficients [tex]\( 5 \)[/tex] and [tex]\( -4 \)[/tex]:
[tex]\[
5 \times (-4) = -20
\][/tex]
### Step 2: Multiplying the Powers of [tex]\( x \)[/tex]
For the variable [tex]\( x \)[/tex]:
Using the laws of exponents, we add the exponents when multiplying like bases:
[tex]\[
x^7 \times x^4 = x^{7+4} = x^{11}
\][/tex]
### Step 3: Multiplying the Powers of [tex]\( y \)[/tex]
For the variable [tex]\( y \)[/tex]:
Similarly, we add the exponents for [tex]\( y \)[/tex]:
[tex]\[
y^2 \times y^5 = y^{2+5} = y^7
\][/tex]
### Step 4: Combining the Results
Now we combine the results from steps 1 through 3:
[tex]\[
(5 x^7 y^2)(-4 x^4 y^5) = -20 x^{11} y^7
\][/tex]
This simplified expression on the left-hand side is:
[tex]\[
-20 x^{11} y^7
\][/tex]
### Step 5: Comparing with the Right-hand Side
Given the right-hand side of the equation is:
[tex]\[
-20 x^8 y^b
\][/tex]
For the equations to be equal, the exponents and coefficients must match on both sides. Thus, we compare:
[tex]\[
-20 x^{11} y^7 = -20 x^8 y^b
\][/tex]
By comparing the exponents of [tex]\( x \)[/tex]:
[tex]\[
x^{11} = x^a \Rightarrow a = 11
\][/tex]
By comparing the exponents of [tex]\( y \)[/tex]:
[tex]\[
y^7 = y^b \Rightarrow b = 7
\][/tex]
### Conclusion
Thus, the values that satisfy the equation are [tex]\( a = 11 \)[/tex] and [tex]\( b = 7 \)[/tex].
Among the given choices, the correct one is:
[tex]\[
\boxed{a=11, b=7}
\][/tex]
