Certainly! Let’s expand the given expression step by step.
The expression we need to expand is:
[tex]\[
(-x + 4) \left(3x^2 - 2x - 7\right)
\][/tex]
To expand this expression, we will use the distributive property, also known as the FOIL method (First, Outer, Inner, Last), to multiply each term in the first binomial by each term in the second trinomial.
1. Distribute [tex]\( -x \)[/tex]:
[tex]\[
-x \cdot \left(3x^2 - 2x - 7\right) = (-x) \cdot (3x^2) + (-x) \cdot (-2x) + (-x) \cdot (-7)
\][/tex]
[tex]\[
= -3x^3 + 2x^2 + 7x
\][/tex]
2. Distribute [tex]\( 4 \)[/tex]:
[tex]\[
4 \cdot \left(3x^2 - 2x - 7\right) = 4 \cdot (3x^2) + 4 \cdot (-2x) + 4 \cdot (-7)
\][/tex]
[tex]\[
= 12x^2 - 8x - 28
\][/tex]
3. Combine the results of both distributions:
Combine like terms (terms with the same power of [tex]\( x \)[/tex]):
[tex]\[
(-3x^3 + 2x^2 + 7x) + (12x^2 - 8x - 28)
\][/tex]
[tex]\[
= -3x^3 + (2x^2 + 12x^2) + (7x - 8x) - 28
\][/tex]
[tex]\[
= -3x^3 + 14x^2 - x - 28
\][/tex]
Thus, the expanded expression in its standard polynomial form is:
[tex]\[
-3x^3 + 14x^2 - x - 28
\][/tex]
That's the polynomial in its standard form.
